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Theorem eldm1cossres2 34553
Description: Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
Assertion
Ref Expression
eldm1cossres2 (𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉

Proof of Theorem eldm1cossres2
StepHypRef Expression
1 eldm1cossres 34552 . 2 (𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝑥𝑅𝐵))
2 elecALTV 34373 . . . 4 ((𝑥 ∈ V ∧ 𝐵𝑉) → (𝐵 ∈ [𝑥]𝑅𝑥𝑅𝐵))
32el2v1 34330 . . 3 (𝐵𝑉 → (𝐵 ∈ [𝑥]𝑅𝑥𝑅𝐵))
43rexbidv 3200 . 2 (𝐵𝑉 → (∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅 ↔ ∃𝑥𝐴 𝑥𝑅𝐵))
51, 4bitr4d 271 1 (𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145  wrex 3062  Vcvv 3351   class class class wbr 4786  dom cdm 5249  cres 5251  [cec 7894  ccoss 34315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7898  df-coss 34511
This theorem is referenced by: (None)
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